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9. Descent Graph Methods
and vice versa. Two subtle points of this process are worth stressing. After
swapping subtrees, we have paternally derived genes flowing to maternal
nodes and vice versa. The obvious adjustments must be made in the chil-
dren and grand-children to correct these forbidden patterns of gene flow.
Also, if either parent has children with another spouse, then that parent’s
relevant subtrees are reduced. Only the paths descending through the chil-
dren shared with the chosen spouse are pertinent. Problem 11 asks readers
to provide a formal description of the sequence of T 0 and T 1 transitions
invoked in executing a T 2a or T 2b transition.
1/1 2/2 1/1 2/2 1/1 2/2 1/1 2/2

3/3 3/3 3/3 3/3

Graph A Graph C

1/2 1/2

1/1 2/2 1/1 2/2

3/3 3/3

1/2
Graph B
FIGURE 9.6. Failure of Descent Graphs A and C to Communicate
One of the complications in constructing a Markov chain on legal de-
scent graphs is that two states may not communicate in the presence of
three or more alleles per marker. Figure 9.6 gives a counterexample in-
volving a single marker locus. In the pedigree depicted in Figure 9.6, all
founders are typed and homozygous; the great-grandchild is typed and het-
erozygous. This great-grandchild must receive his allele 1 from one pair of
great-grandparents and his allele 2 from the other pair. The two possibilities
are labeled descent graph A and descent graph C. However, it is impossible
to move in a finite number of transitions from descent graph A to descent